Free Resolutions of Parameter Ideals for Some Rings with Finite Local Cohomology

FREE RESOLUTIONS OF PARAMETER IDEALS FOR SOME RINGS WITH FINITE LOCAL COHOMOLOGY

HAMID RAHMATI

Abstract. Let R be a d-dimensional local ring, with maximal ideal m, con- taining a ﬁeld and let x1, . . . , xd be a system of parameters for R. If depth R ≥ d−1 d − 1 and the local cohomology module Hm (R) is ﬁnitely generated, then i i there exists an integer n such that the modules R/(x1, . . . , xd) have the same Betti numbers, for all i ≥ n.

Introduction Throughout this note R is a local noetherian ring with maximal ideal m. Let M be a finitely generated R-module and let F be a minimal free resolution of R P∞ i M. The Poincaréseries of M is the formal power series PM (t) = i=0(rank Fi)t . i F We let ΩR(M) denote the ith syzygy of M, that is to say Coker ∂i+1. Let x = n x1, . . . , xd be a system of parameters for R. For each n ∈ N, let x denote the n n sequence x1 , . . . , xd . Several classical results in local algebra establish that ideals contained in the large powers of the maximal ideal exhibit a similar behavior. In his thesis, [7], Y. H. Lai considers the following question, which he attributes to D. Katz: Do the Poincaréseries of the modules R/(xn) behave uniformly, for n large enough? Clearly, the answer is yes if R is Cohen-Macaulay: In this case xn is a regular sequence, for each n ≥ 1, so R/(xn) is resolved by a Koszul complex on d elements. It is also not difficult to obtain a positive answer when dim R = 1. Lai proves that if dim R = 2, depth R = 1 and the local cohomology module 1 Hm(R) is finitely generated, then for n large enough one has R 2 2 R P n (t) = 1 + 2t + t + t P 1 (t). R/(x ) Hm(R) Now we state our main result. It evidently covers the first two cases mentioned above. Also part (ii) of our main theorem generalizes Lai’s result, since the Canon- ical Element Conjecture holds for 2-dimensional rings. Main Theorem. Let R be d-dimensional local ring with maximal ideal m. If d−1 d − depth R ≤ 1 and the R-module H = Hm (R) is finitely generated, then there exists an integer n, such that for each system of parameters x for R contained in mn the following assertions hold d+1 ∼ d−1 (i) ΩR (R/(x)) = ΩR (H). (ii) If in addition, the Canonical Element Conjecture holds for R, then one has

R d 2 R PR/(x)(t) = (1 + t) + t PH (t).

1991 Mathematics Subject Classiﬁcation. Primary 13D02, 13D40; Secondary 13H10. 1 2 HAMID RAHMATI

The Cohen-Macaulay defect of R is the number cmd R = dim R − depth R. One always has cmd R ≥ 0, and equality characterizes Cohen-Macaulay rings. On the i other hand, Cohen-Macaulay rings are also characterized by the equality Hm(R) = 0 for all i 6= d. Comparing these conditions with the hypotheses on R in the theorem, one may say that these hypotheses, in some sense, give the least possible extension of Cohen-Macaulay rings.

1. Free resolutions of almost complete intersection ideals

Let y = y1, . . . , yn be a sequence in R. We let K(y; M) denote the Koszul complex on y with coeﬃcients in M. Set

Hi(y; M) = Hi(K(y; M)). P∞ i P∞ i If P (t) = i=0 ait and Q(t) = i=0 bit are formal power, we write P 4 Q to indicate that ai ≤ bi holds for all i ≥ 0. The main result of this section is the following theorem: Theorem 1.1. Let R be a d-dimensional ring and let x be a system of parameters for R. Set Hi = Hi(x; R) for i = 1, . . . , d. One then has cmd R X PR (t) (1 + t)d + ti+1 PR (t). R/(x) 4 Hi i=1 Moreover, if cmd R ≤ 1 and the Canonical Element Conjecture holds for R, then one has PR (t) = (1 + t)d + t2 PR (t). R/(x) H1 Under the hypotheses of the second part of the theorem, x generates an almost complete intersection ideal. Some of the discussion below is carried out in this more general framework. Recall that an ideal I is called almost complete intersection if grade I ≥ µ(I) − 1, where grade I is the maximal length of an R-regular sequence in I and µ(I) is the number of minimal generators of I. Now we give an example that shows the inequality in Theorem 1.1 can be strict if depth R < d − 1. Example 1.2. Set R = k[[a, b, c]]/(ac, bc, c2), then dim R = 2 and depth R = 0. Consider the system of parameters x = a, b. Using Macaulay 2 we get: P R (t) = 1 + 3t + 6t2 + 13t3 + 28t4 + ··· H2 P R (t) = 3 + 7t + 12t2 + 26t3 + 56t4 + ··· H1 R 2 3 4 PR/(x)(t) = 1 + 2t + 3t + 7t + 15t + ··· . Thus one has R 2 3 4 PR/(x)(t) ≺ 1 + 2t + 4t + 8t + 15t + ··· = (1 + t)2 + t2P R (t) + t3P R (t). H1 H2 Let X be a complex of R-modules and let ∂X denote its diﬀerential, set

sup H(X) = sup{n ∈ Z | Hn(X) 6= 0}. Let Σt denote the shift functor deﬁned by t ΣX t X (Σ X)n = Xn−t and ∂n = (−1) ∂n−t. FREE RESOLUTIONS OF PARAMETER IDEALS 3

Let α: X → Y be a morphism of complexes. Recall that mapping cone of α is deﬁned to be the complex C such that Cn = Xn−1 ⊕ Yn and C X Y ∂n ((x, y)) = (−∂n−1(x), ∂n (y) + αn−1(x)) for all (x, y) ∈ Cn. A quasi-isomorphism is a morphism of complexes that induces isomorphism in homology in all degrees. Lemma 1.3. Let X be a complex with s = sup H(X) < ∞ and let F be a free s resolution of Hs(X). There exists a morphism of complexes α:Σ F → X such that the mapping cone X0 of α satisﬁes ( 0 ∼ 0 i ≥ s Hi(X ) = Hi(X) i ≤ s − 1.

Proof. Let τ≥s(X) be the complex X · · · −→ Xs+2 −→ Xs+1 −→ ker ∂s −→ 0 and ι: τ≥s(X) → X be the inclusion map. Since F is a bounded below complex of free modules, there is a morphism of complexes β : F → τ≥s(X) such that the following diagram is commutative

β s ι Σ F / τ≥s(X) / X JJ JJ JJ π s JJ Σ JJ% s Σ Hs(X), where : F → Hs(X) and π are quasiisomorphism. Set α = ιβ and let C be the mapping cone of α. One has an exact sequence 0 −→ X −→ X0 −→ Σs+1F −→ 0 of complexes. It induces an exact sequence of homology modules

0 s+1 Hi(α) · · · → Hi+1(X) → Hi+1(X ) → Hi+1(Σ F ) −−−−→ Hi(X) → · · · .

From the construction of α one sees that Hs(α) is an isomorphism, and since s+1 ∼ Hi(Σ F ) = Hi−s−1(F ) = 0 for all i 6= s + 1, we get the desired result.

Remark 1.4. Assume X is a complex of free modules such that Xi = 0, for i < 0, i and Hi(X) = 0, for i > s, where s is some positive integer. Let F be a free resolution of Hi(X), for i = 1, . . . , s. Applying Lemma 1.3 s times, one gets a free 1 s resolution G of H0(X) such that Gi = Xi ⊕ Fi−2 ⊕ · · · ⊕ Fi−s−1. However, even if the complexes X,F 1,...,F s are minimal, G need not be minimal; see Example 1.2. Now we show the relation between the Poincaréseries of an almost complete intersection ideal and the Poincaréseries of its first Koszul homology module.

Lemma 1.5. Let I be an almost intersection ideal of R and let y = y1, . . . , yr be a minimal set generators for I. Set H = H1(y; R). One then has r+1 r−1 ΩR (R/I) = ΩR (H) R r+1 R PR/I (t) = Q(t) + t PN (t), r−1 where Q(t) is a polynomial of degree r and N = ΩR (H). 4 HAMID RAHMATI

Proof. Let K denote the Koszul complex of y with coeﬃcients in R. Since grade I ≥ r − 1, one has Hi(y; R) = 0 for i 6= 0, 1. Let F be a minimal free resolution of H. Let α:ΣF → K be the map from Lemma 1.3 and let C be the mapping cone of α. The complex C is a complex of free modules and has only one nonvanishing ∼ ∼ homology module namely, H0(C) = H0(y; R) = R/I, so C is a free resolution of R/I. By construction of mapping cones the complex C is minimal if and only if αF ⊆ mK. Since F is minimal and Cm = Fm for all m ≥ r + 1, non-minimality can only happen in the ﬁrst r + 1 degrees, therefore

r+1 r ΩR (R/I) = ΩR(H). This completes the proof of the ﬁrst equality. The second equality follows from the ﬁrst one.

1.6. For a system of parameters x for R, let Fx be a free resolution of R/(x) and let γx : Kx = K(x; R) → Fx be a lifting of the map Kx → R/(x). The following are equivalent: (i) The Canonical Element Conjecture holds for R. (ii) For every system of parameters x for R and every free resolution Fx of R/(x), the map H(k ⊗R γx): H(k ⊗R Kx) → H(k ⊗R Fx) is injective. It is shown in [1, (1.6)] that the equivalence of (i) and (ii) follows from a theorem of P. Roberts [8]. A proof of Roberts’ theorem is given in [6, (1.3)]. The Canonical Element Conjecture holds for R provided R is equicharacteristic or dim R ≤ 3: Hochster has proved that the Canonical Element Conjecture is equivalent to the Direct Summand Conjecture, and that the conjectures hold if R is equicharacteristic or dim R ≤ 2, see [5]. In [3], R. Heitmann shows that the Direct Summand Conjecture, hence the Canonical Element Conjecture, holds for every 3-dimensional ring. Proof of Theorem 1.1. The inequality follows from Remark 1.4. For the rest of the proof, we keep the notation in the proof of Lemma 1.5. To prove the equality it suﬃces to show that C is minimal. Assume not, and let K F 0 ≤ n ≤ d − 1 be such that ∂(Cn) * mCn−1. Since Im(∂ ) and Im(∂ ) are in mC, there exists an element f such that ∂(f) = e ∈ Kn\mKn. It follows that f is not in mFn hence Rf is a direct summand of Cn, and Re is a direct summand of Kn. Let G be the complex 0 → Rf −→λ Re → 0 where Re is in degree n and λ is the C 0 restriction of ∂n . Set C = C/G. Since λ is an isomorphism, X is exact, hence is a free resolution of R/(x). Let π : C → C be the natural surjection. The inclusion map ι: K → C is a lifting of the augmentation map α: K → R/(x). Thus πι is a lifting of α, but Hn(k ⊗R πι)(1 ⊗ e) = 0. This contradicts, by 1.6, the hypothesis that Canonical Element Conjecture holds for R. Therefore, C is minimal; this implies the theorem.

2. standard system of parameters Let M be a finitely generated R-module. A system of parameters x for M is said to be standard if i (x)Hm(M/(x1, . . . , xj)M) = 0 FREE RESOLUTIONS OF PARAMETER IDEALS 5 holds, for all non-negative integers i, j with i + j < d. For information about standard system of parameters we refer the reader to [10] and [11]. An R-module M is said to have finite local cohomology if for each integer i ≤ i dim M − 1 the local cohomology module Hm(M) is of finite length. Modules with finite local cohomology are also called generalized Cohen-Macaulay modules. The following statement is a consequence of [11, (2.1) and (3.1)] and [9, (3.7)]. 2.1. An R-module M has finite local cohomology if and only if there exists a positive integer n such that every system of parameters in mn is standard.

i For an R-module M we set H (y; M) = H−i(HomR(K(y; R),M)). One then has ∼ d−i Hi(y; M) = H (y; M), for all i, see [2, 1.6.10]. Theorem 2.2. Let M have finite local cohomology. Set g = depth M. If x is a standard system of parameters for M, then for every integer n ≥ 1, the canonical g n g map λn :H (x ; M) → Hm(M) is an isomorphism. To prove this theorem we need to recall some facts about standard system of parameters and modules with finite local cohomology. The next statement follows from [11, (3.1)] and [9, (3.3)]. 2.3. Let x be a system of parameters for M. If M has finite local cohomology and depth M = g, then the subsequence x1, . . . , xg of x is M-regular. In [4, (1)], standard systems of parameters are characterized in terms of Koszul homology: 2.4. Assume M has finite local cohomology. Let x be a system of parameters for M. The following are then equivalent: (i) x is standard. 2 2 (ii) `(Hp(x1, . . . , xr; M)) = `(Hp(x1, . . . , xr; M)), for p ≥ 1 and 1 ≤ r ≤ d. Pr−p r i (iii) `(Hp(x1, . . . , xr; M)) = i=0 i+p `(Hm(M)), for all p ≥ 1 and 1 ≤ r ≤ d. In particular, if x is standard, then g `(Hd−g(x; M)) = `(Hm(M)), where g = depth M and d = dim M. Here `(M) denotes the length of M. The next result is [4, (4)].

2.5. Let M have ﬁnite local cohomology, and let x1, . . . , xd be a system of parameters for M. Let n1, . . . , nd be positive integers. For all p > 0 and 1 ≤ r ≤ d and for all positive integers m1, . . . , mr satisfying n1 ≤ m1, . . . , nr ≤ mr, one has

n1 nr m1 m1 `(Hp(x1 , . . . , xr ; M)) ≤ `(Hp(x1 , . . . , xr ; M)). Lemma 2.6. Let M have ﬁnite local cohomology and let x be a standard system of parameters for M. For all positive integers p and m, one has

m `(Hp(x; M)) = `(Hp(x ; M)). Proof. For every i > 0, from 2.4, one obtains: 2i `(Hp(x; M)) = `(Hp(x ; M)). 6 HAMID RAHMATI

For each integer m > 0 there exists an integer i > 0 such that 2i ≥ m. Using 2.5, we get inequalities

m 2i `(Hp(x; M)) ≤ `(Hp(x ; M) ≤ `(Hp(x ; M)), which imply the desired statement.

Next we recall the relation between Koszul homology and local cohomology . Let (n) n (n) (n) x be a system of parameters for M and set K = K(x ; R) and let e1 , . . . , ed denote the standard basis of K(n) ' Rd. For all n ≥ 1, there is a commutative diagram

ϕ(n) (n+1) 1 / (n) K1 K1

(n+1) (n) K0 K0

(n) (n+1) (n) where ϕ1 (ej ) = xjej for j = 1, . . . , d. This deﬁnes a morphism of complexes (n) (n) (n+1) (n) (n) (n) (n) ϕ = ∧ϕ1 : K → K . Set ψ = HomR(ϕ ,M), then ψ induces a i i n i (n+1) map α(n) :H (x ; M) → H (x ; M). By [2, 3.5.6], one has Hi (M) = lim Hi(xn; R) for all i ≥ 0. m −→

Lemma 2.7. Let z = z1, ··· , zt be a sequence in R and let s be the length of maximal M-regular sequences in (z). For each positive integer n , if y = y1, ··· , yr is an M-regular sequence in (zn+1), then one has the following commutative diagram

s α(n) Hs(zn; M) / Hs(zn+1; M) O O ' ' s−r α¯(n) Hs−r(zn; M) / Hs−r(zn+1; M) where M = M/(y)M. Proof. If r = 1, then one has a short exact sequence

y1 0 −→ M −→ M −→ M/y1M −→ 0. It induces a commutative diagram

s−1 y s−1 n ðn s n 1 s n 0 / H (z ; M/y1M) / H (z ; M) / H (z ; M)

s−1 αs αs α(n) (n) (n) s−1 ðn+1 y s−1 n+1 s n+1 1 s n+1 0 / H (z ; M/y1M) / H (z ; M) / H (z ; M).

g n g n+1 g−1 Since y1 annihilates both H (z ; M) and H (z ; M), the connecting maps ðn g−1 and ðn+1 are isomorphisms and this gives the desired commutative diagram for r = 1. The general case follows by iteration. FREE RESOLUTIONS OF PARAMETER IDEALS 7

0 Proof of Theorem 2.2. Let n ≥ 1. The sequence x = x1, . . . , xg is M-regular, see 2.3. So x0n+1 is an M-regular sequence in the ideals (xn+1) ⊆ (xn). Set M = M/(x0)M. By 2.7, we get the following commutative diagram

g α(n) Hg(xn; M) / Hg(xn+1; M) O O ' ' 0 α¯(n) H0(xn; M) / H0(xn+1; M).

0 The mapα ¯(n) is injective, since it is induced by the identity map in the following commutative diagram (n) / / d / HomR(K , M) = 0 M M ···

ψ(n) = (n+1) / / d / HomR(K , M) = 0 M M ··· . g Therefore α(n) is injective. g n g n+1 g The modules H (x ; M) and H (x ; M) have the same length, see 2.6, so α(n) is an isomorphism and this implies the desired statement.

3. free resolutions of parameter ideals In this section we assume that R is a d-dimensional ring with ﬁnite local cohomology, and cmd R ≤ 1. Because of the property recalled in 2.1, the following result contains the Main Theorem stated in the introduction. Theorem 3.1. Let x be a standard system of parameters for R and set H = d−1 Hm (R). One then has d+1 ∼ d−1 (i) ΩR (R/(x)) = ΩR (H). (ii) If in addition, the Canonical Element Conjecture holds for R, then one has R d 2 R PR/(x)(t) = (1 + t) + t PH (t). Proof. One has grade(x) = depth R and µ(x) = dim R, so grade(x) ≥ µ(x) − 1, then part (i) follows from Lemma 1.5 and Theorem 2.2, and part (ii) follows from Theorem 1.1 and Theorem 2.2 .

Acknowledgments I should like to express my gratitude to my advisers L. L. Avramov and S. Iyengar for their support and guidance throughout the development of this paper.

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[3] R. C. Heitmann, The direct summand conjecture in dimension 3, Ann. of Math. (2) 156 (2002), no. 2, 695-712. [4] L. T. Hoa, Koszul homology and generalized Cohen-Macaulay modules, Acta Math. Vietnam 18 (1993), no. 1, 91-98. [5] M. Hochster, Canonical elements in local cohomology and the direct summand conjecture, J. Algebra 84 (1983), 503-553 [6] C. Huneke and J. Koh, Some dimension 3 cases of the canonical element conjecture., Proc. Amer. Math. Soc. 98 (1986), no. 3, 394-398. [7] Y. H. Lai, On the relation type of system of parameters and on the Poincar´eseries of system of parameters., Ph.D Thesis Purdue Unviversity, 1995 [8] P. Roberts, The equivalence of two forms of the Canonical Element Conjecture, undated manuscript. [9] P. Schenzel, N. V. Trung and N. T. Coung, Verallgemeinerte Cohen-Macaulay Moduln, Math Nachr, 85 (1978), 57-73. [10] J. St¨uckard and W. Vogel, Buchsbaum rings and applications, Berlin-Heidelberg-New York, Springer-Verlog, 1986. [11] N. V. Trung, Toward a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102 (1986), 1-49.

Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A. E-mail address: [email protected]